# Optionals IAS Mains Mathematics: Questions 204 - 210 of 283

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## Question 204

Appeared in Year: *2016*

### Describe in Detail

Essay▾Find the locus of the point of intersection of three mutually perpendicular tangent planes to the Conicoid (CS Paper -1)

### Explanation

The Equation of the central Conicoid be

… eq. (1)

Let the equation of one of the three mutually tangent planes to the central Conicoid (1) be

Where being the actual direction – cosines and

i.e.. … eq. (2)

Similarly let the equation of the other two tangent planes be

… eq. (3)

… eq. (4)

Squaring and adding, are get

Which in the required locus

… (5 more words) …

## Question 205

Appeared in Year: *2008*

### Describe in Detail

Essay▾Reduce to canonical form. (CS paper -2)

### Explanation

Given differential equation is

It can be written as

… eq. (1)

Compare (1) with we have

Therefore, equation gives

Let

Therefore, equation become

Integrating we get, … eq. (2)

Let

And

Putting values of r and t in (1) , we get

Which is the required canonical form of given equation.

… (2 more words) …

## Question 206

Appeared in Year: *2012*

### Describe in Detail

Essay▾Solve the partial differential equation (CS paper -2)

### Explanation

Given, Partial Differential Equation is

Where are arbitrary functions

The required solution is

>

## Question 207

Appeared in Year: *2013*

### Describe in Detail

Essay▾What is the maximal possible order of an element in ? Give an example of such an element. How many element՚s will there be in of that order? (Paper II)

### Explanation

**NOTE: -** For convenience, we denote an n-cycle by then all the possible disjoint cycle structures of element of according to longest cycle lengths left to right, we have

Now, we know that order of a permutation of finite set written in disjoint cycle form is the least common multiple of the lengths of the cycles.

The possible order of the elements …

… (35 more words) …

## Question 208

Appeared in Year: *2011*

### Describe in Detail

Essay▾Show that the best of six transformation on set complex numbers defined by

Is a non-abelian group of order 6 w. r. t. composition of mappings. (Paper II)

### Explanation

- Here
T. P is non-abelian group

- The composition table is given below.
Table Shows the Element in Composition 0 - (i) Closure Property:
Since all the element in composition table are elements of G, so G is closed under multiplication.

- (ii) Associativity:
Since the transformation composition is always associative

So G composition is also associative in G also.

- …

… (101 more words) …

## Question 209

Appeared in Year: *2007*

### Describe in Detail

Essay▾Let T be the linear transformation from defined by

For each

Determine a basic for the range space of T. what is the dimension of the range space of T? (Paper I)

### Explanation

- We have defined by
- We know that standard basis of is
Now

- We know that range space of T is spanned by , therefore to find a basis of range space of T, we find the L. I. vectors from
- Consider a matrix A whose rows are generators of range of T and reduce it to echelon from
is a basis of

- Now, let
… eq. (1)

- Now, to find the basis of null space of T is e…

… (53 more words) …

## Question 210

Appeared in Year: *2008*

### Describe in Detail

Essay▾Determine the maximum and minimum distances of the origin from the curve given by the (Paper I)

### Explanation

- Let be any point on the surface
… eq. (1)

Let

- Where is the distance of from
Let

- Where is Lagrange multiplier
- For extreme points,
Or … eq. (2)

And … eq. (3)

- Since x and y are both non-zero
**Take****Take**- Hence maximum value of is and minimum value is .

… (3 more words) …